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Theorem xmulval 10813
Description: Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmulval  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  (
( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  = 
-oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) ) ) )

Proof of Theorem xmulval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 445 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
21eqeq1d 2446 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  0  <-> 
A  =  0 ) )
3 simpr 449 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
43eqeq1d 2446 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  =  0  <-> 
B  =  0 ) )
52, 4orbi12d 692 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  =  0  \/  y  =  0 )  <->  ( A  =  0  \/  B  =  0 ) ) )
63breq2d 4226 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 0  <  y  <->  0  <  B ) )
71eqeq1d 2446 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  +oo  <->  A  =  +oo ) )
86, 7anbi12d 693 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( 0  < 
y  /\  x  =  +oo )  <->  ( 0  < 
B  /\  A  =  +oo ) ) )
93breq1d 4224 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  <  0  <->  B  <  0 ) )
101eqeq1d 2446 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  -oo  <->  A  =  -oo ) )
119, 10anbi12d 693 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( y  <  0  /\  x  = 
-oo )  <->  ( B  <  0  /\  A  = 
-oo ) ) )
128, 11orbi12d 692 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( 0  <  y  /\  x  =  +oo )  \/  (
y  <  0  /\  x  =  -oo ) )  <-> 
( ( 0  < 
B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) ) ) )
131breq2d 4226 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 0  <  x  <->  0  <  A ) )
143eqeq1d 2446 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  =  +oo  <->  B  =  +oo ) )
1513, 14anbi12d 693 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( 0  < 
x  /\  y  =  +oo )  <->  ( 0  < 
A  /\  B  =  +oo ) ) )
161breq1d 4224 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <  0  <->  A  <  0 ) )
173eqeq1d 2446 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  =  -oo  <->  B  =  -oo ) )
1816, 17anbi12d 693 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  <  0  /\  y  = 
-oo )  <->  ( A  <  0  /\  B  = 
-oo ) ) )
1915, 18orbi12d 692 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( 0  <  x  /\  y  =  +oo )  \/  (
x  <  0  /\  y  =  -oo ) )  <-> 
( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) ) )
2012, 19orbi12d 692 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( ( 0  <  y  /\  x  =  +oo )  \/  ( y  <  0  /\  x  =  -oo ) )  \/  (
( 0  <  x  /\  y  =  +oo )  \/  ( x  <  0  /\  y  = 
-oo ) ) )  <-> 
( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) ) ) )
216, 10anbi12d 693 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( 0  < 
y  /\  x  =  -oo )  <->  ( 0  < 
B  /\  A  =  -oo ) ) )
229, 7anbi12d 693 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( y  <  0  /\  x  = 
+oo )  <->  ( B  <  0  /\  A  = 
+oo ) ) )
2321, 22orbi12d 692 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( 0  <  y  /\  x  =  -oo )  \/  (
y  <  0  /\  x  =  +oo ) )  <-> 
( ( 0  < 
B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) ) ) )
2413, 17anbi12d 693 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( 0  < 
x  /\  y  =  -oo )  <->  ( 0  < 
A  /\  B  =  -oo ) ) )
2516, 14anbi12d 693 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  <  0  /\  y  = 
+oo )  <->  ( A  <  0  /\  B  = 
+oo ) ) )
2624, 25orbi12d 692 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( 0  <  x  /\  y  =  -oo )  \/  (
x  <  0  /\  y  =  +oo ) )  <-> 
( ( 0  < 
A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo ) ) ) )
2723, 26orbi12d 692 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( ( 0  <  y  /\  x  =  -oo )  \/  ( y  <  0  /\  x  =  +oo ) )  \/  (
( 0  <  x  /\  y  =  -oo )  \/  ( x  <  0  /\  y  = 
+oo ) ) )  <-> 
( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  ( ( 0  < 
A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo ) ) ) ) )
28 oveq12 6092 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  x.  y
)  =  ( A  x.  B ) )
2927, 28ifbieq2d 3761 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( ( ( 0  <  y  /\  x  =  -oo )  \/  ( y  <  0  /\  x  = 
+oo ) )  \/  ( ( 0  < 
x  /\  y  =  -oo )  \/  (
x  <  0  /\  y  =  +oo ) ) ) ,  -oo , 
( x  x.  y
) )  =  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) )
3020, 29ifbieq2d 3761 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( ( ( 0  <  y  /\  x  =  +oo )  \/  ( y  <  0  /\  x  = 
-oo ) )  \/  ( ( 0  < 
x  /\  y  =  +oo )  \/  (
x  <  0  /\  y  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  y  /\  x  =  -oo )  \/  ( y  <  0  /\  x  =  +oo ) )  \/  (
( 0  <  x  /\  y  =  -oo )  \/  ( x  <  0  /\  y  = 
+oo ) ) ) ,  -oo ,  ( x  x.  y ) ) )  =  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  (
( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  = 
-oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) ) )
315, 30ifbieq2d 3761 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( x  =  0  \/  y  =  0 ) ,  0 ,  if ( ( ( ( 0  <  y  /\  x  =  +oo )  \/  (
y  <  0  /\  x  =  -oo ) )  \/  ( ( 0  <  x  /\  y  =  +oo )  \/  (
x  <  0  /\  y  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  y  /\  x  =  -oo )  \/  ( y  <  0  /\  x  =  +oo ) )  \/  (
( 0  <  x  /\  y  =  -oo )  \/  ( x  <  0  /\  y  = 
+oo ) ) ) ,  -oo ,  ( x  x.  y ) ) ) )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) ) ) )
32 df-xmul 10714 . 2  |-  x e  =  ( x  e. 
RR* ,  y  e.  RR*  |->  if ( ( x  =  0  \/  y  =  0 ) ,  0 ,  if ( ( ( ( 0  <  y  /\  x  =  +oo )  \/  (
y  <  0  /\  x  =  -oo ) )  \/  ( ( 0  <  x  /\  y  =  +oo )  \/  (
x  <  0  /\  y  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  y  /\  x  =  -oo )  \/  ( y  <  0  /\  x  =  +oo ) )  \/  (
( 0  <  x  /\  y  =  -oo )  \/  ( x  <  0  /\  y  = 
+oo ) ) ) ,  -oo ,  ( x  x.  y ) ) ) ) )
33 c0ex 9087 . . 3  |-  0  e.  _V
34 pnfxr 10715 . . . . 5  |-  +oo  e.  RR*
3534elexi 2967 . . . 4  |-  +oo  e.  _V
36 mnfxr 10716 . . . . . 6  |-  -oo  e.  RR*
3736elexi 2967 . . . . 5  |-  -oo  e.  _V
38 ovex 6108 . . . . 5  |-  ( A  x.  B )  e. 
_V
3937, 38ifex 3799 . . . 4  |-  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  ( ( 0  < 
A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo ) ) ) ,  -oo ,  ( A  x.  B ) )  e.  _V
4035, 39ifex 3799 . . 3  |-  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) )  e.  _V
4133, 40ifex 3799 . 2  |-  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  = 
-oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) ) )  e. 
_V
4231, 32, 41ovmpt2a 6206 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  (
( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  = 
-oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   ifcif 3741   class class class wbr 4214  (class class class)co 6083   0cc0 8992    x. cmul 8997    +oocpnf 9119    -oocmnf 9120   RR*cxr 9121    < clt 9122   x ecxmu 10711
This theorem is referenced by:  xmulcom  10847  xmul01  10848  xmulneg1  10850  rexmul  10852  xmulpnf1  10855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-mulcl 9054  ax-i2m1 9060
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-pnf 9124  df-mnf 9125  df-xr 9126  df-xmul 10714
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